The Truncated Exponential Polynomials, the Associated Hermite Forms and Applications

We discuss the properties of the truncated exponential polynomials and develop the theory of new form of Hermite polynomials, which can be constructed using the truncated exponential as a generating function. We derive their explicit forms and comment on their usefulness in applications, with particular reference to the theory of flattened beams, used in optics.


Introduction
In a previous paper, Dattoli et al. [5] have discussed the properties of families of Hermite polynomials, defined through the generating function (g.f.) (1.1) The function f (x), called the support function, is assumed to be infinitely differentiable, to admit the series expansion and the "semigroup" property (1.2b) The operator function f (y) is defined as with f r + acting on f s in such a way that f r + f s = f s+r .
(1.2d)Under these assumptions, the polynomials Φ n (x, y) can explicitly be written as In this paper, we will discuss a particular case of the polynomials (1.3), having as support the polynomials of the truncated exponential (TEP), namely, where x r r! , (1.5) are the PTE, according to the definition of [1].The reason of the interest for this family of polynomials stems from the fact that they currently appear in the theory of the so-called flattened beams, which plays a role of paramount importance in optics and in particular in the case of super-Gaussian optical resonators (see, e.g., [6]).
A comprehensive theory of the TEP has been developed in [4], where different generalizations have been proposed and we will take some advantages from that general treatment.
The PTE can also be defined by means of the g.f.
satisfying the recurrences which can be combined to obtain the relevant second-order differential equation [1] xe n − (n + x)e n + ne n = 0. (1.7) For future convenience, we will write the PTE in the form G. Dattoli and M. Migliorati 3 where θ(x) denotes the Heaviside step function, defined in such a way that (1.9) By comparing (1.2) and (1.8), it is evident that in the case of PET f r = θ(n − r).We can therefore obtain the explicit form of the Hermite polynomials of the truncated exponential (THP), as it will be shown in the forthcoming section.

Hermite polynomials of the truncated exponential
According to the previous discussion, the THP can explicitly be written as and we can use (1.5) and (1.6) to establish the relevant recurrences.By keeping the derivatives of both sides of (1.5) with respect to x, y, t, we obtain three recurrences, involving the continuous variables and the discrete indices, namely, ( We can combine the first two recurrences to infer that the HPTE satisfy the "heat" type equation where f + is an operator, defined in such a way that (2.4) (A kind of ambiguity seems to arise in the definition of the operators f ± .Strictly speaking, according to the definition given in the previous section, we should have and the operator should therefore act on the indices where the summation series is running.We prefer, for practical reasons, the definition given in (2.4), which induces discrete variations on the index m; it is however evident that f ± = f ∓ .)We can handle (2.3) to obtain the following operational definition for the TEP: a proper combination of the last recurrence in (2.2) with the first yielding the identity which, together with the first recurrence of (2.2), leads to the following second-order differential equation (see [5]): (According to the formalism of monomiality (see [4]), we can define a multiplicative operator (in this case provided by M m ) and a derivative operator P = ∂/∂x; the THPs are easily seen to satisfy the identity Needless to say, all the previous relations reduce to those of the ordinary Hermite for ( Note that H n (x,−1/2) can be identified with the polynomials He n (x), defined by the generating function exp(xt − t 2 /2).)Let us now consider the problem of finding a simple formula yielding the successive derivatives, with respect to t, of the right-hand side of (1.5).Using the previous definitions, we get where H (x, y | m) should be understood as an operator, defined as follows: (2.9) The proof of the previous relations can be achieved in a fairly direct way.By multiplying, indeed, the right-hand side of (2.8) by ξ p / p! and then summing over p, we get (2.10) and thus, using the semigroup property (1.2b), we easily end up with (2.8).In alternative, (2.8) can also be written as (2.11) The results obtained in this section provide the backbone of the topics we will discuss in the forthcoming sections.

Some applications of TEP and THP
The flattened beams have been introduced in [6] to study optical systems employing the so-called super-Gaussian beams, namely, optical beams whose transverse shape is not G. Dattoli and M. Migliorati 5 reproduced by a simple Gaussian, but by a function exhibiting a quasi constant flat top as, for example, Unlike the ordinary Gaussian beams, the super-Gaussians do not have transparent propagation properties, which can easily be exploited in the design of an optical resonator.The flattened beams defined as provide a good tool of approximation of a super-Gaussian, and their paraxial evolution can be treated using straightforward analytical tools.
A super-Gaussian can be reproduced according to the relation S(x, p) ∼ = F(ax,m), with a,m conveniently chosen in correspondence of the order p of the super-Gaussian.An example is shown in Figure 3.1, where we have made the comparison between F(4.76x,20) and a super-Gaussian of order p = 10.
In Figure 3.2, we also report a comparison between a flattened beam function (FBF) and an ordinary Gaussian.
A first useful information characterizing the FBF can be obtained by deriving the differential equation they satisfy.By noting indeed that we find from (1.7) that which explicitly yield the fairly simple form (note that H n (2x,−1) = H n (x) = n! [n/2]  r=0 ((−1) r (2x) n−2r /(n − 2r)!r!) are the ordinary Hermite defined by the g.f.exp(2xt − t 2 )), we obtain Let us now discuss how the above formalism can be exploited to treat problems associated with the propagation of an FBF.

G. Dattoli and M. Migliorati 7
We therefore consider the following heat-type equation: whose formal solution can be written as To get an explicit solution, we should recall some rules of operational nature.The use of the well-known identity (3.12) making therefore a combined use of the modified Burchnall formula (see, e.g., [3]) of the identity (some times referred to as the Gleisher operational rule (see [8])) and of the following properties of the Hermite polynomials: we end up with where we have defined which is a Hermite-based polynomial of the truncated exponential, whose properties have been studied in [4].An example of evolution at different times of the Y (x,t) function is given in Figure 3.3.Analogous results can be obtained for the case of the free paraxial (Schrödinger) equation whose solution is omitted for the sake of conciseness.

Concluding remarks
Before closing the paper, let us add some comments aimed at better framing the obtained results.
We therefore go back to the definition of FBF and consider the evaluation of average quantities like To perform such a calculation, we consider the following integral: G. Dattoli and M. Migliorati 9 The explicit evaluation of this integral can be done using the generating function method (1.7), thus we get where are two index Hermite polynomials (see [2,3]).It is easily checked that in the case of (4.3) the odd indices are zero.It is therefore evident that the FBF root mean square at different times can be written as which can be exploited to study the behavior of transverse section of a flattened beam.The function can be considered a kind of truncated Gaussian and can be exploited in the theory of approximation; it shares interesting properties with the ordinary Gaussian and indeed we get where We have stressed that the use of FBF simplifies the problems associated with the evaluation of beam transport, which in the case of super-Gaussian beams should be evaluated numerically.We have shown that analytical solutions can be obtained using operational methods and the properties of the TEP.Different methods can however be adopted.As also indicated in [6], the use of the expansion of the TEP in terms of Laguerre polynomials allows the use of the properties of Laguerre-Gauss optical beams, whose propagation properties are well documented in the literature (see [7]) and the method we have proposed is just an alternative to this early suggestion.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: